Clearly not every variety has the Hilbert property: for example, in [1]} it is shown that – as a consequence of the Chevalley-Weil theorem – a smooth proper variety over a number field with the Hilbert property is geometrically simply-connected.
Nonetheless, the literature is by now very rich with examples of varieties with the Hilbert property. Several proofs exist for the case of projective space (see for example [2]}, [3]}, [4]}, [5]}, [6]}, [7]}). Note that connected reductive algebraic groups have been shown in [8]} to have the Hilbert property over Hilbertian fields; notice that these varieties are rational over \(\overline{k}\) , but not necessarily over \(k\) ). Some sporadic examples of varieties with the Hilbert property are also known (such as the K3 surface \(x^4+y^4 = z^4+w^4 \subset \mathbb {P}_{3,\mathbb {Q}}\) , which is not even geometrically unirational [1]}; see also [10]} and [11]}).
